Question

Given: ∆ABC, CD ⊥ AB, AC = BC, Area of ABC = 32 cm2, m∠A = 72°

Find: CD

Answer 1

9.92 cm

Explanation:

Using Trig functions, we can calculate the lengths.

Answer 2

9.92 cm

Explanation:

Given: ∆ABC,

CD ⊥ AB,

AC = BC,

Area of ABC = 32 cm2,

m∠A = 72°

Find: CD

Solution:

Triangle ABC is isosceles triangle with base AB, because AC = BC. In isosceles triangle angles adjacent to the base are congruent. So,

`m\angle A=m\angle B=72^(\circ)`

The sum of the measures of all interior angles is 180°, thus

`m\angle C=180^(\circ)-m\angle A-m\angle B\\ \\m\angle C=180^(\circ)-72^(\circ)-72^(\circ)=36^(\circ)`

The area of the triangle ABC is

`A_(ABC)=(1)/(2)AC\cdot BC\cdot \sin \angle C\\ \\32=(1)/(2)AC^2\sin 36^(\circ)\\ \\AC^2=(64)/(\sin 36^(\circ))\\ \\AC\approx 10.43\ cm`

Consider right triangle ACD. In this triangle,

`\sin \angle A=(CD)/(AC)\\ \\\sin 72^(\circ)=(CD)/(10.43)\\ \\CD=10.43\sin 72^(\circ)\approx 9.92\ cm`